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GILBERT STRANG: OK.

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More about eigenvalues
and eigenvectors.

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Well, actually, it's
going to be the same thing

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about eigenvalues
and eigenvectors

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but I'm going to
use matrix notation.

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So, you remember I have a
matrix A, 2 by 2 for example.

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It's got two eigenvectors.

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Each eigenvector
has its eigenvalue.

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So I could write the
eigenvalue world that way.

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I want to write
it in matrix form.

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I want to create an
eigenvector matrix

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by taking the two
eigenvectors and putting them

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in the columns of my matrix.

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If I have n of them, that
allows me to give one name.

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The eigenvector matrix, maybe
I'll call it V for vectors.

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So that's A times V. And
now, just bear with me

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while I do that
multiplication of A times

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the eigenvector matrix.

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So what do I get?

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I get a matrix.

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That's 2 by 2.

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That's 2 by 2.

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You get a 2 by 2 matrix.

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What's the first column?

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The first column of
the output is A times

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the first column of the input.

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And what is A times x1?

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Well, A times x1 is
lambda 1 times x1.

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So that first column
is lambda 1 x1.

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And A times the second column
is Ax2, which is lambda 2 x2.

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So I'm seeing lambda
2 x2 in that column.

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OK.

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Matrix notation.

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Those were the eigenvectors.

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This is the result of A times V.
But I can look at this a little

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differently.

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I can say, wait a minute, that
is my eigenvector matrix, x1

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and x2-- those two
columns-- times a matrix.

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Yes.

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Taking this first column, lambda
1 x1, is lambda 1 times x1,

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plus 0 times x2.

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Right there I did a
matrix multiplication.

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I did it without
preparing you for it.

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I'll go back and do that
preparation in a moment.

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But when I multiply a matrix by
a vector, I take lambda 1 times

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that one, 0 times that one.

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I get lambda 1 x1,
which is what I want.

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Can you see what I want
in the second column here?

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The result I want
is lambda 2 x2.

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So I want no x1's, and
lambda 2 of that column.

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So that's 0 times that
column, plus lambda 2,

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times that column.

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Are we OK?

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So, what do I have now?

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I have the whole thing
in a beautiful form,

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as this A times the
eigenvector matrix

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equals, there is the
eigenvector matrix again, V.

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And here is a new matrix
that's the eigenvalue matrix.

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And everybody calls
that-- because those

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are lambda 1 and lambda 2.

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So the natural letter
is a capital lambda.

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That's a capital Greek lambda
there, the best I could do.

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So do you see that the two
equations written separately,

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or the four equations
or the n equations,

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combine into one
matrix equation.

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This is the same as
those two together.

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Good.

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But now that I have
it in matrix form,

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I can mess around with it.

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I can multiply both
sides by V inverse.

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If I multiply both sides by
V inverse I discover-- well,

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shall I multiply on
the left by V inverse?

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Yes, I'll do that.

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If I multiply on the left by
V inverse that's V inverse AV.

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This is matrix multiplication
and my next video

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is going to recap
matrix multiplication.

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So I multiply both
sides by V inverse.

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V inverse times V
is the identity.

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That's what the
inverse matrix is.

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V inverse, V is the identity.

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So there you go.

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Let me push that up.

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That's really nice.

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That's really nice.

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That's called diagonalizing
A. I diagonalize

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A by taking the eigenvector
matrix on the right,

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its inverse on the left,
multiply those three matrices,

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and I get this diagonal matrix.

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This is the diagonal
matrix lambda.

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Or other times I might want
to multiply by both sides

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here by V inverse
coming on the right.

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So that would give me A, V,
V inverse is the identity.

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So I can move V over
there as V inverse.

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That's what it amounts to.

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I multiply both
sides by V inverse.

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So this is just A and this is
the V, and the lambda, and now

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the V inverse.

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That's great.

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So that's a way to see how
A is built up or broken down

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into the eigenvector matrix,
times the eigenvalue matrix,

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times the inverse of
the eigenvector matrix.

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OK.

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Let me just use
that for a moment.

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Just so you see how it
connects with what we already

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know about eigenvalues
and eigenvectors.

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OK.

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So I'll copy that great fact,
that A is V lambda, V inverse.

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Oh, what do I want to do?

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I want to look at A squared.

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So if I look at
A squared, that's

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V lambda V inverse
times another one.

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Right?

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There's an A, there's an
A. So that's A squared.

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Well, you may say I've made
a mess out of A squared,

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but not true.

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V inverse V is the identity.

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So that it's just the identity
sitting in the middle.

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So the V at the far left,
then I have the lambda,

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and then I have the other
lambda-- lambda squared--

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and then the V inverse
at the far right.

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That's A squared.

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And if I did it n
times, I would have

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A to the n-th what would be
the lambda to the n-th power V

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inverse.

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What is this?

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What is this saying about?

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This is A squared.

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How do I understand
that equation?

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To me that says that the
eigenvalues of A squared

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are lambda squared.

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I'm just squaring
each eigenvalue.

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And the eigenvectors?

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What are the eigenvectors
of A squared?

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They're the same V, the
same vectors, x1, x2,

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that went into v. They're
also the eigenvectors

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of A squared, of A cubed, of
A to the n-th, of A inverse.

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So that's the point of
diagonalizing a matrix?

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Diagonalizing a
matrix is another way

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to see that when I square
the matrix, which is usually

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a big mess, looking at the
eigenvalues and eigenvectors

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it's the opposite of a big mess.

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It's very clear.

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The eigenvectors are
the same as for A.

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And the eigenvalues are squares
of the eigenvalues of A.

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In other words, we can
take the n-th power

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and we have a nice
notation for it.

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We learned already
that the n-th power

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has the eigenvalues
to the n-th power,

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and the eigenvectors the same.

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But now I just see it here.

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And there it is
for the n-th power.

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So if I took the same
matrix step 1,000 times,

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what would be important?

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What controls the thousandth
power of a matrix?

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The eigenvectors stay.

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They're just set.

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It would be the thousandth
power of the eigenvalue.

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So if this is a matrix with
an eigenvalue larger than 1,

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then the thousandth
power is going

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to be much larger than one.

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If this is a matrix with
eigenvalues smaller than 1,

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there are going to
be very small when

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I take the thousandth power.

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If there's an eigenvalue
that's exactly 1,

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that will be a steady state.

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And 1 to the thousandth
power will still be 1

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and nothing will change.

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So, the stability.

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What happens as I multiply,
take powers of a matrix,

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is a basic question parallel
to the question what

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happens with a
differential equation

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when I solve forward in time?

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I think of those two
problems as quite parallel.

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This is taking steps, single
steps, discrete steps.

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The differential equation is
moving forward continuously.

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This is a difference
between hop,

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hop, hop in the discrete case
and run forward continuously

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in the differential case.

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In both cases, the eigenvectors
and the eigenvalues

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are the guide to what
happens as time goes forward.

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OK.

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I have to do more about
working with matrices.

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Let me come to that next.

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Thanks.